The fourth property is called "positive definiteness". Typically when moving over to other fields one replaces this with "non-degeneracy" (which is weaker than positive definiteness).
4'. $\langle v,w \rangle=0$ for all $w$ implies that $v=0$.
Note: If you move to working over the complex numbers then usually one replaces 3. with conjugate symmetry. And so this along with 2. imply that pulling scalars out of the second slot results in conjugation.
There are indefinite scalar product spaces. I suggest reading "Indefinite Linear Algebra and Applications" by Gohberg, Lancaster, Rodman. Applications are wide; to name a few: theory of relativity and the research of polarized light (mostly Minkowski space is used here), and matrix polynomials (nicely covered in "Matrix polynomials", again by Gohberg, Lancaster, Rodman).
Each indefinite scalar product space is induced by a nonsingular Hermitian indefinite matrix $J$ by
$$[x, y] := \langle Jx, y \rangle = y^* J x,$$
although I've seen other variations as is usuall with this (for example, $x^* J y$).
The vectors which you've mentioned, $x \ne 0$, but $[x,x] = 0$, are usually called degenerate, although other names are used as well, for example neutral. In later terminology (I think used mostly by physicists), $x$ for which $[x,x] < 0$ is called negative, and if $[x,x] > 0$, $x$ is called positive.
The most common indefinite scalar product space is hypperbolic, induced by $J = \mathop{\rm diag}(j_1,\dots,j_n)$, where $j_k \in \{-1,1\}$. Minkowski space is usually defined by $j_1 = \pm 1$ and $j_k = \mp 1$ for $k > 1$, or $j_k = \pm 1$ for $k < n$ and $j_n = \mp 1$.
There are wider classes of scalar products on finite real and complex spaces (I think there was even some work on spaces of quaternions) than indefinite ones. For example, orthosymmetric products, for which $J$ need not be Hermitian, but $J^* = \tau J$ for some $\tau \in \mathbb{C}$ such that $|\tau| = 1$.
Another widely researched class are symplectic scalar products, induced by $J = \left[\begin{smallmatrix} & {\rm I}_n \\ -{\rm I}_n \end{smallmatrix}\right]$.
I suggest reading Tisseur, especially her "Structured Factorizations in Scalar Product Spaces, Higham, Sanja Singer (especially "Orthosymmetric block reflectors" with Saša Singer), Mehrmann, Mehl, maybe few of my own papers,...
Best Answer
Assume this really defines an inner product $\langle\ ,\ \rangle$.
Consider $u=(1,0)$ and $v=(0,1)$. Then the angle between $w=u+v$ and $u$ and the angle between $w$ and $v$ are both $\alpha=\pi/4$ and $\cos(2\alpha)=0$ hence $\langle u,w\rangle=\langle v,w\rangle=0$, which implies that $\langle w,w\rangle=\langle u,w\rangle+\langle v,w\rangle=0$.
This is a contradiction because $\|w\|\ne0$ and the angle that $w$ makes with $w$ is $\beta=0$, and $\cos(2\beta)\ne0$, hence $\langle w,w\rangle=\|w\|\cdot\|w\|\cdot\cos(2\beta)=\|w\|^2\ne0$.